Atlas › Modular forms › Weight 2 › Level 704 › 704.2.a › 704.2.a.o

Newform orbit 704.2.a.o

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + q^{3} - 3q^{5} + 3q^{9} + 2q^{11} - 4q^{13} + 7q^{15} + 6q^{17} + 6q^{19} + 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43} + 13T_{47} + 14T_{53} + 15T_{59} + 16T_{61}$ , whose minimal polynomial is

$x^{2} - 220x - 80992$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 704$ )
11	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	0	1	−3	0	2	−4	6	6	3	−6	15	−1	0	6	8	0	−13	6	3	−19	−12	18	2	−3	−1

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.815 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 176, 352, 704]: [0, 1, 2, 3, 2, 4, 5, 6, 9, 10, 16, 28, 52, 100] (certified at ell = 2147482801)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12), (47, 13), (53, 14), (59, 15), (61, 16)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(704, 2)[14]   # 704.2.a.o

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 704:2 1000